Energy Stable Boundary Conditions for the Nonlinear Incompressible Navier-Stokes Equations

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1 Energy Stable Boundary Conditions for the Nonlinear Incompressible Navier-Stokes Equations Jan Nordström, Cristina La Cognata Department of Mathematics, Computational Mathematics, Linköping University, Linköping, SE , Sweden Abstract The nonlinear incompressible Navier-Stokes equations with boundary conditions at far fields and solid walls is considered. Two different formulations of boundary conditions are derived using the energy method. Both formulations are implemented in both strong and weak form and lead to an estimate of the velocity field. Equipped with energy bounding boundary conditions, the problem is approximated by using difference operators on summation-by-parts form and weak boundary and initial conditions. By mimicking the continuous analysis, the resulting semi-discrete as well as fully discrete scheme are shown to be provably stable, divergence free and high-order accurate. Keywords: Navier-Stokes equations, incompressible, boundary conditions, energy estimate, stability, summation-by-parts, high-order accuracy, divergence free 1. Introduction The nonlinear incompressible Navier-Stokes equations are regularly used in models of climate and weather forecasts, ocean circulation predictions [23, 35, 42], studies of turbulent airflow around vehicles [13, 19], studies of blood flow [7, 41], analysis of pollution [10, 33] and many others fields. Various formulations of the incompressible Navier-Stokes model have been proposed. The velocity-pressure formulation, where the explicit divergence addresses: (Jan Nordström), (Cristina La Cognata)

2 relation is omitted, is the most common choice. Popular numerical techniques to enforce zero divergence for this form include staggered grids [9, 21] and projections or fractional step methods [38, 18]. Yet another procedure is to modify the pressure equation [11, 34] or devise boundary conditions [17, 30] which systematically damp the divergence inside the computational domain. In this paper we consider the Navier-Stokes equations in the original velocity-divergence form directly, which bypasses the need for the special divergence reducing techniques mentioned above. The boundary conditions are derived in a form that is similar to the characteristic boundary conditions for hyperbolic problems (and generalized to the compressible Navier-Stokes equations in [25, 32]). We follow the general procedure for initial boundary values problems (IBVP) outlined in [26] and define outgoing and ingoing variables at the boundaries. The latter are specified in terms of the former and data in order to get an energy estimate. Two formulations stemming from two different techniques to diagonalize the boundary terms are presented. In the first formulation, the boundary conditions are obtained through a non-singular rotation while, in the second formulation, they are derived directly by an eigenvalue decomposition. Both formulations are strongly and weakly imposed and we observe that they naturally come in nonlinear form. We derive general boundary conditions, but pay particular attention to the specific ones at a solid wall. Furthermore, it is shown that it is not necessary to provide pressure data in the initial and boundary conditions to obtain the full solution. The nonlinear system is discretized in space and time by using difference operators on Summation-By-Parts (SBP) form [40, 37, 24]. The boundary and initial conditions are weakly imposed with the Simultaneous-Approximation-Term (SAT) technique [4, 39]. The resulting SBP-SAT approximation is proved to be stable in both a semi-discrete and fully discrete sense. The derivation is done for the finite difference version of SBP-SAT, but it is equally valid for other approximations such as finite volume [28, 27], spectral elements [3, 2], discontinuous Galerkin [12, 8] and flux reconstruction schemes [15, 5] on SBP-SAT form. The paper proceeds as follows. In Section 2, we introduce and discuss the continuous problem and derive boundary conditions. Next, in Section 3, the general form of boundary conditions are specified to fit a solid wall. In Section 4 is shown how to impose the boundary condition without involving the pressure. A comparison discussing the advantages and disadvantages of the two formulations at far field and solid walls is provided in Section 5. In 2

3 Section 6, the semi-discrete version of the governing equations and the SAT terms for the boundary conditions are derived, and stability is proven. The fully discrete SBP-SAT approximation and a complete stability analysis is presented in Section 7. Conclusions are drawn in Section The continuous problem Consider the incompressible Navier-Stokes equations with velocity field u = (u, v), pressure p and viscosity ɛ u t + uu x + vu y + p x ɛ (u xx + u yy ) = 0, v t + uv x + vv y + p y ɛ (v xx + v yy ) = 0, (1) u x + v y = 0. By letting v = (u, v, p) T and introducing the matrices u 0 1 v A = 0 u 0, B = 0 v 1 and Ĩ = 0 1 0, (2) the system (1) can be written as Ĩv t + Av x + Bv y ɛĩ (v xx + v yy ) = 0. (3) Next, we rewrite the convection terms in (1) as Av x = 1 2 (Av) x Av x 1 2 A xv, Bv y = 1 2 (Bv) y Bv y 1 2 B yv. (4) By inserting (4) into (3) and recalling that we are dealing with an incompressible fluid, i.e., A x + B y = (u x + v y )Ĩ = 0, we obtain the initial boundary value problem for the skew-symmetric form of the incompressible Navier- Stokes equations Ĩv t [(Av) x + Av x + (Bv) y + Bv y ] ɛĩ v = 0, (x, y) Ω, t > 0 (5) Hv = g, (x, y) Ω, t > 0 (6) v = f, (x, y) Ω, t = 0. (7) In (6)-(7), g and f are the initial and boundary data, respectively. The boundary operator H will be specified on Ω such that the correct (minimal) number and type of boundary conditions are imposed. 3

4 Remark 2.1. The nonlinear terms must be split into skew-symmetric form for the upcoming discrete analysis. For more details regarding different splitting techniques, see [16, 6, 36]. Note that the systems (3) and (5) are symmetric which allows for a straightforward use of the energy method. Remark 2.2. Existence requires that (6) constitutes a minimal set of boundary conditions. This means that the correct (minimal) number of linearly independent rows in H must imposed. Too few boundary conditions will neither lead to an estimate nor uniqueness. Remark 2.3. We consider smooth compatible data and, consequently, smooth solutions for the problem (5)-(7). Normally, nonlinear well-posedness would follow as an extension of linear well-posedness through the linearization and localisation principles, see [20]. However, no explicit bound on the pressure is derived in this paper, hence we do not refer to the resulting formulation as well-posed The energy estimate The energy method and Green s theorem applied to (5) yield d dt v 2 + Ĩ 2ɛ v 2 = BT, (8) Ĩ where v 2 = Ĩ Ω vt Ĩv is a semi-norm that allows for v Ĩ = 0 even for p 0. In (8), BT denotes the boundary term BT = v T (An x + Bn y ) v 2ɛv T Ĩ [v x n x + v y n y ] ds, (9) Ω where ds = dx 2 + dy 2 and n = (n x, n y ) is the outward pointing unit normal vector on Ω. To derive an estimate based on (8), we must bound BT. We start by introducing a change of variables u n u s w = p ɛ n u n = T v, T = ɛ n u s n x n y 0 n y n x ɛ n n x ɛ n n y 0, (10) ɛ n n y ɛ n n x 0 4

5 where u n and u s denote the outward normal and tangential velocity components, respectively, and n = n = n x x + n y y is the normal derivative. Next, we apply (10) to (9) and rearrange such that BT becomes BT = Ω u n u s p ɛ n u n ɛ n u s u n u n 0 u n u s p ɛ n u n } {{ 0 0 } ɛ n u s A n T ds. (11) We need a minimal number of boundary conditions such that w T A n w Boundary conditions We follow the roadmap in [26] for IBVP s and focus on the items: 1. The number of boundary conditions. The boundary term (11) will be diagonalized using different techniques. The minimal number of boundary conditions is equal to the number of negative diagonal entries. 2. The form of the boundary conditions. The transformed variables associated to the negative diagonal elements (ingoing variables) are specified in terms of the ones corresponding to positive diagonal elements (outgoing variables) and boundary data. 3. The strong implementation. The boundary conditions are chosen such that a negative semi-definite boundary term is obtained for zero boundary data. 4. The weak implementation. The weak imposition of the new boundary conditions is chosen such that it leads to the same estimate as the strong imposition augmented with a dissipative boundary term The number of boundary conditions using rotations A symmetric matrix A can be diagonalized by a suitable non-singular rotation matrix M. The elements of the resulting diagonal matrix, Λ = M T AM, are not necessarily the eigenvalues of A, but the number of positive and negative diagonal entries is the same. For more details regarding diagonalizations with rotations see [32, 26]. 5

6 A complete diagonalization of the boundary matrix A n in (11) is given by 1 u n u n Λ M = M T A n M, with rotation M = (12) The diagonal matrix and vector of linearly independent rotated variables are u 2 n + p ɛ n u n Λ M = u n and W = M u n u s ɛ n u s 1 w = ɛ n u n p ɛ n u n ɛ n u s (13) Note that, the matrix Λ M in (13) always has two positive and two negative diagonal entries. Consequently, the problem (5) requires two boundary conditions both at an inflow boundary (u n < 0) and an outflow boundary (u n > 0). Next, we write Λ M = diag(λ +, 0, Λ ), where Λ and Λ + contain the negative and positive diagonal elements, and indicate with W and W + the corresponding variables. In the inflow case we have [ u W 2 = n + p ɛ n u n u n u s ɛ n u s ], W + = [ p ɛ n u n ɛ n u s ], Λ = 1 u n I 2, Λ + = Λ, (14) while in the outflow case, the signs are flipped and we have p W ɛ n u = n u, W + 2 = n + p ɛ n u n, Λ = 1 I ɛ n u s u n u s ɛ n u 2, Λ + = Λ. s u n (15) In both situations, the variable corresponding to the zero eigenvalue is W 0 = ɛ n u n, while W and W + are called the in- and outgoing rotated variables, respectively. With this notation, the quadratic form in (11) can be written [ W + W + BT = W ] T [ Λ Λ W ], (16) where the variable corresponding to the zero diagonal entry is ignored. 6

7 The number of boundary conditions using eigenvalues The correct number of boundary conditions can also be obtained by directly finding the eigenvalues of A n in (9), see also [31, 30]. The eigenvalue problem A n λi 5 = 0 defines a characteristic polynomial equation with five distinct real roots λ 1 < λ 2 < λ 3 < λ 4 < λ 5 and eigenvalue matrix Λ = diag(λ 1,..., λ 5 ), where λ 1,5 = u n 2 (un 2 ) 2 + 2, λ3 = 0, λ 2,4 = u n 2 (un ) (17) The associated orthonormal basis of eigenvectors is indicated by X = XN and it leads to the eigenvalue decomposition λ λ 5 A n = XΛX T 0 λ 2 0 λ 4 0 = XNΛ(XN) T, where X = (18) and N 1 = diag( 2 + λ 2 1, 1 + λ 2 2, 2, 1 + λ 2 3, 2 + λ 2 5) contains the normalizing weights of the columns in X. The diagonal matrix and linearly independent characteristic variables are λ 1 u n + p ɛ n u n λ 2 u s ɛ n u s Λ X = NΛN T and W = X T w = p + ɛ n u n (19) λ 4 u s ɛ n u s λ 5 u n + p ɛ n u n respectively. Next, we introduce W + = (X T w) + and W = (X T w) given by W λ1 u = n + p ɛ n u n, W + λ4 u = s ɛ n u s. (20) λ 2 u s ɛ n u s λ 5 u n + p ɛ n u n With a slight abuse of notation, we denote by W and W + the in- and outgoing characteristic variables, respectively. The variable corresponding to the zero eigenvalue is W 0 = p + ɛ n u n. Note that λ 1, λ 2 < 0 and λ 4, λ 5 > 0 for all values of u n. This implies that (as in the previous case) we need two boundary conditions. Moreover, given 2 7

8 (20) and the diagonal matrices Λ λ1 /(2 + λ = 2 1) 0 0 λ 2 /(1 + λ 2, Λ + λ4 /(1 + λ = 2 4) 0 2) 0 λ 5 /(2 + λ 2 5) (21) we can again rewrite BT in (11) in the diagonal form (16). Remark 2.4. The number of boundary conditions is independent of the specific transformation used to arrive at the diagonal form (16), as long as the resulting variables are linearly independent. This follows from Sylvester s low of inertia, see [32, 14] for details. The two specific transformations presented above (there might be even more) lead to different forms of boundary conditions, which is the next topic The form of the boundary conditions We start by proving Proposition 1. The form of boundary condition that bounds (16) cannot involve the variable corresponding to the zero eigenvalue. Proof. It suffices to consider the homogeneous case of the form W = RW + + R 0 W 0 and inserting it in (16). We find W + T Λ BT = + + R T Λ R R T Λ R 0 W + Ω W 0 (R 0 ) T Λ R (R 0 ) T Λ R 0 W 0 ds, which directly implies that R 0 must be identical to zero. The general form of boundary conditions that bounds the right-hand side in (16) (as well as in (9)) is hence given by Hv = W RW + = g, (22) where R is a 2-by-2 matrix and g = (g 1, g 2 ) T contains the boundary data. This is a nonlinear version of the result in [26] for a linear IBVPs. The formulation (22) decomposes the boundary operator H into Hv = (H RH + )v. (23) In the rotated formulation, H + and H are given by H + v = (M 1 ) + T v = W +, H v = (M 1 ) T v = W, (24) 8

9 with M 1 from (12) and T from (10). In particular, for the inflow case (M 1 ) + un =, (M 1 ) = (25) 0 u n while in outflow case they are interchanged. In a similar way, the characteristic variable formulation gives H + v = (X T ) + T v = W +, H v = (X T ) T v = W, (26) with X from (18) and (X T ) λ =, (X T ) + = 0 λ λ (27) λ Remark 2.5. The boundary conditions in (22)-(27) are in general nonlinear The implementation procedure We start with the strong form of the boundary procedure The strong implementation The following proposition is a nonlinear version of the linear result in [26]. Proposition 2. The boundary conditions (22) bounds (16) if Λ + + R T Λ R > 0 (28) and a positive semi-definite or positive definite matrix Γ exists such that Λ + (Λ R)[Λ + + R T Λ R] 1 (Λ R) T Γ <. (29) Proof. Consider condition (22) and replace W with RW + + g in (16) to get W + T Λ BT = + + R T Λ R R T Λ W + Ω g Λ R Λ ds. (30) g By adding and subtracting Ω gt Γg ds to (30), where Γ is a positive semidefinite or positive definite matrix, we find W + T Λ BT = + + R T Λ R R T Λ W + Ω g Λ R Γ + Λ ds + g 9 Ω g T Γg ds. (31)

10 Now consider the following matrix decomposition Λ + + R T Λ R R T Λ Λ R Γ + Λ = Y T MY, Y = I Z, 0 I where Z = [Λ + + R T Λ R] 1 R T and Λ M = + + R T Λ R 0 0 Γ + Λ (Λ R)[Λ + + R T Λ R] 1 (Λ R) T. (32) If (28) holds and we choose Γ such that the lower bound in (29) holds, it follows that the matrix M in (32) is positive semi-definite and (31) becomes W + T W BT = Y T + MY ds + g T Γg ds g T Γg ds. (33) Ω g g Ω Ω If in addition, also the upper bound in (29) holds, we have an estimate. Remark 2.6. In the linear case studied in [26], condition (28) suffices. Corollary 1. The homogeneous boundary conditions W = RW + leads to a bound for (16) if Λ + + R T Λ R 0. (34) Proof. Consider (22) with g = 0. The boundary term (30) becomes BT = W [ +T Λ + + R T Λ R ] W + ds 0. (35) Ω The weak implementation The boundary conditions (22) can also be weakly imposed by adding the penalty term L(Σ(W RW + g)) to the right-hand side of (5) yielding Ĩv t [(Av) x + Av x + (Bv) y + Bv y ] ɛĩ v = L(Σ(W RW + g)). (36) Here, L is a lifting operator [1] defined such that, for smooth vector functions φ, ψ φ T L(ψ) dxdy = φ T ψ ds, Ω Ω holds. In (36), Σ is a penalty matrix to be determined. 10

11 Proposition 3. The weak imposition of the boundary conditions in (36) with leads to an energy estimate if (28) and (29) holds. Σ = (H ) T Λ (37) Proof. The energy method applied to (36) leads to (8) with two additional boundary terms BT = + Ω Ω W + T Λ + 0 W + W 0 Λ W ds v T Σ(W RW + g) + [ v T Σ(W RW + g) ] T ds. (38) The introduction of Σ in (37) such that v T Σ = (W ) T Λ leads to The splitting BT = Ω W + W g Λ + R T Λ 0 W + Λ R Λ Λ W ds. (39) 0 Λ 0 g T Λ + R T Λ 0 R T Λ R R T Λ R T Λ Λ R Λ Λ = Λ R Λ Λ 0 Λ 0 Λ R Λ Λ Λ + + R T Λ R 0 R T Λ (40) Λ R 0 Γ + Λ 0 0 Γ transforms (39) to BT = Ω Ω (W RW + g) T Λ (W RW + g) ds W + T Λ + + R T Λ R R T Λ W + g Λ R Γ + Λ ds + g Ω g T Γg ds. (41) Clearly, the first term on the right-hand side of (41) is non-positive. The other two are identical to the ones in (31) obtained with the strong imposition and lead to (33). 11

12 Corollary 2. The weak imposition of the homogeneous boundary condition in (36) together with (37) leads to an estimate if (34) holds. Proof. Consider the homogeneous version of (36), i.e., with data g = 0 and Σ as in (37). The same procedure as in the proof of Proposition3 leads to W + T Λ + R BT = T Λ W + W Λ R Λ W ds. Ω By adding and subtracting (W + ) T [R T Λ R]W + ds, we find BT = (W + ) T [Λ + + R T Λ R]W + + [W RW + ] T Λ [W RW + ] Ω which is non-positive if (34) holds. Remark 2.7. The weak imposition produces the same energy rate as the strong imposition with an additional damping term. A similar term will appear in the discrete approximation and stabilize it The continuous energy estimate We have proved that if condition (28) and (29) required in Proposition 2 and 3 hold, the boundary condition (22) yield Ω d dt v 2 Ĩ + 2ɛ v 2 Ĩ g T Γgds. (42) Time integration of (42) yields the final energy estimate ( T T v 2Ĩ)t=T + 2ɛ v 2 Ĩ f 2 + g T Γg. (43) Ĩ 0 Remark 2.8. The relation (43) bounds the velocity field only. Note also that no initial condition on the pressure is required. 3. Solid wall boundary conditions For a solid wall, the specific form (22) is most easily derived by seeking matrices R and S such that W RW + un = S. (44) Relation (44) defines a system of equations for the elements in S and R. If a solution exists, the solid wall conditions are obtained by imposing (44) with zero right-hand side. 12 u s 0

13 3.1. Solid wall rotated boundary conditions In the rotated formulation, the sign of u n changes the form of W ± in (44). Consider the inflow case for u n 0 and the variables in (14). The system (44) has the solution We can prove R = and S = u n. (45) 0 1 Proposition 4. The inflow solid wall boundary conditions (44) with R in (45) leads to an energy bound. Proof. Consider the boundary conditions in (44) with zero right-hand side. As proved in Corollary 1 and 2, condition (34) must be satisfied in order to get an energy bound. From (14) and (45), it follows that Λ + + R T Λ R = u n [ 1 ] T u n = Next, we examine the outflow case for u n 0+ and consider (15). Now the system defined by (44) has the solution R = and S = u 0 1 n, (46) 0 1 We conclude with the following result, similar to Proposition 4. Proposition 5. The outflow solid wall boundary conditions (44) with R in (46) leads to an energy bound. Proof. See the proof of Proposition Solid wall characteristic boundary conditions In this formulation, a unique expression for the ingoing and outgoing variables is given by (20). By solving for S and R in (44) we find 0 1 d1 0 R = and S =, (47) d 2 where d 1 = λ 1 λ 5 = u 2 n + 8 and d 2 = λ 2 λ 4 = u 2 n + 4. We prove 13

14 Proposition 6. The characteristic form of the boundary conditions at a solid wall (44) with R in (47) leads to an energy bound. Proof. From (21) and (47), it follows that condition (34) holds since [ λ4 ] [ 0 T λ1 ] [0 ] Λ + +R T Λ (1+λ R = 2 4 ) (2+λ ) 1 = λ 5 (2+λ 2 5 ) λ 2 (1+λ 2 2 ) External data requirements for the pressure As stated in Remark 2.8, no initial condition for the pressure is required for the bound (43). Therefore, it is of interest to investigate if it is possible to derive matrices R in (22) which remove the pressure also from the boundary procedure and still obtain an energy estimate. Consider boundary conditions (22) with the inflow rotated variables in (14). The set of matrices that removes the pressure from the boundary conditions is 1 r12 R =, (48) 0 r 22 which yields W RW + = [ u 2 n r 12 ɛ n u s, u n u s (1 + r 22 ɛ n u s ) ] T. The coefficients r 12 and r 22 must be determined such that Proposition 2 and 3 (non-homogeneous case) or Corollary 1 and 2 (homogeneous case) hold. From (14) and (48), we find Λ + + R T Λ R = 1 0 r12 u n r r r22 2. (49) By choosing r 12 = 0 and r 22 1, (50) Λ + + R T Λ R has one zero and one non-negative eigenvalue and condition (34) holds. Similarly, in the outflow case, the same matrix as in (48) yields W RW + = [ u 2 n + r 12 ɛ n u s, u n u s + (1 + r 22 ɛ n u s ) ] T. As in the inflow case, condition (50) implies that (34) is satisfied. Remark 4.1. Note that (50) includes the inflow R in (45) and the outflow R in (46) derived for the solid wall case in Section

15 Consider the characteristic variables in (20). The set of matrices r11 1 R = r 21 0 yields W RW + = [ u n (λ 1 λ 5 ) r 11 (λ 4 u s ɛ n u s ), u n (λ 2 r 21 λ 4 ) + r 21 ɛ n u s ] T which does not involves the pressure. From (14) and (48), it follows that ] By choosing Λ + + R T Λ R = [ λ4 + (1+λ 2 4 ) r2 λ (1+λ 2 2 ) r2 λ 1 λ 11 r 1 (2+λ 2 1 ) 11 (2+λ 2 1 ) r 11 λ 1 (2+λ 2 1 ) 0 (51). (52) r 11 = 0 and r 12 1, (53) Λ + + R T Λ R has one zero and one non-negative eigenvalue which implies that condition (34) holds. Remark 4.2. Note that also condition (53) includes R in (47) derived for the solid wall case in Section 3.2. We conclude that both formulations admit homogeneous energy bounding boundary conditions which do not include the pressure. 5. Similarities and differences between the two formulations The two formulations require the same number of boundary conditions and they both lead to an energy estimate in the homogeneous case. However, despite the similarities, there are differences which will be discussed below The rotated boundary conditions The form of the boundary conditions depends on whether there is an inflow or outflow situation at the boundary. This means that the boundary procedure must adapt to the time evolution of the solution. Moreover, since the penalty matrix in (37) and conditions (28), (29) and (34) depend on the solution, care must be taken when the magnitude of the normal velocity assumes large or small values. From (14) and (15), it follows that Λ ± = ±I/ u n. Hence, for u n 0, Λ ±, while for u n, Λ ± 0. This indicates that this formulation might be problematic for u n small. 15

16 Proposition 7. The boundary conditions (22) in the rotated variable formulation bounds (16) if R T R I for u n δ > 0. The bound is obtained for both the strong and the weak imposition. Proof. Consider the diagonal matrices in (14) and (15). The lower bound in (29) becomes Γ [ I + R(I 2 R T R) 1 R T ] / u n. Hence, conditions (28) and (29) in Proposition 2 and 3 are both satisfied. For a weak imposition in the solid wall case, the possible problem with a vanishing normal velocity is removed. To clarify, consider the general penalty term in (36) with Σ from (34) and g = 0. In the inflow case, we get Σ(W RW + ) = (H ) T Λ un S = (H ) T un, u s u s where while in the outflow case where H = Σ(W RW + ) = (H ) T Λ S T, [ un u s ] H un = 0 u n = (H ) T un, u s In both cases, H is bounded as u n 0 ±. Hence, the singularities that would seemingly occur for vanishing u n are eliminated The characteristic boundary conditions The in- and outgoing characteristic boundary conditions have a fixed form independent of the solution. Furthermore, all the eigenvalues in (17) and the corresponding eigenvectors in (18) remain bounded for all values of u n. In particular, condition (29) is automatically satisfied if (28) holds, which proves the following relaxed version of Proposition 2 Proposition 8. The boundary conditions (22) in the characteristic formulation bounds (16) if Λ + + R T Λ R > 0 holds. The bound holds for both the strong and the weak imposition. T. 16

17 Also, in the solid wall case, all the matrices involved in condition (47) are bounded when u n 0 ±, which leads to a well-defined formulation. Remark 5.1. The comparison in Section 5.1 and 5.2 shows that the characteristic formulation is more suitable than the rotated formulation. It is well-defined for all flow cases and have the same form for all values of u n. In the remaining numerical part of the paper, we will limit ourselves to the characteristic formulation of boundary conditions. 6. The semi-discrete approximation To discretize the system (36) in space, we consider an approximation on SBP-SAT form. In order to make the paper self-contained, we provide a brief introduction to the SBP-SAT discretization and recommend [40, 37, 24] for a complete description. As was mentioned in the introduction, we present the technique using high order accurate finite differences, but the derivation is valid for all approximations on SBP-SAT form. To derive a semi-discrete energy estimate, we will mimic the analysis of the continuous case above. Consider a two-dimensional Cartesian grid of N M points with coordinates (x i, y j ). The west, east, south and north boundaries are indicated by b {W,E,S,N} and the normals at each boundary by n b = (n b x, n b y), see Figure 1. y n N = (n N x, nn y ) = (0, 1) n W = (n W x, nw y ) = ( 1, 0) Ω n E = (n E x, ne y ) = (1, 0) n S = (n S x, ns y ) = (0, 1) x Figure 1: Two-dimensional domain showing the outward pointing normals. 17

18 The discrete approximation of a variable v = v(x, y) is a vector of length N M arranged as v = (v 11,..., v 1M, v 21..., v 2M,..., v N1,..., v NM ) T, where v ij v(x i, y j ). The SBP approximation of the spatial partial derivatives of v are given by D x v = (Px 1 Q x I M )v v x and D yv = (I N Py 1 Q y )v v y, where I d is the identity matrix of dimension d and denotes the Kronecker product [14]. The matrices P x,y are diagonal, positive definite and such that the product (P x P y ) forms a quadrature rule which defines a discrete L 2 norm v 2 P x P y = v T (P x P y )v. The operators Q x,y are almost skewsymmetric matrices satisfying the SBP property Q x + Q T x = E 0 + E N, Q y + Q T y = E 0 + E M, (54) where E 0 = diag(1, 0,..., 0) and E N,M = diag(0,..., 0, 1), with the appropriate dimensions. The matrix D(a) has the components of the vector a injected on the diagonal The semi-discrete formulation Consider the time-dependent vector V = (u(t), v(t), p(t)) T and the discrete version of (2) D(u) 0 I NM D(v) 0 0 I NM 0 0 A = 0 D(u) 0, B = 0 D(v) I NM, Ĩ3 = 0 I NM 0. I NM I NM (55) Here, A, B and Ĩ3 are 3NM 3NM matrices while 0 is a NM NM matrix of zeros. With this notation, the semi-discrete SBP-SAT approximation of (36) becomes Ĩ 3 V t + D x V + D ] y V ɛ [(Ĩ D x) 2 + (Ĩ D y) 2 V = Pen BT, (56) where the skew-symmetric splitting (4) is approximated by the difference operators D x = 1 2 (I 3 D x )A+ 1 2 A(I 3 D x ), Dy = 1 2 (I 3 D y )B+ 1 2 B(I 3 D y ). (57) 18

19 In (56), Pen BT is the discrete version of the lifting operator in (36). It can be decomposed into the sum of four terms corresponding to each boundary of the square domain, namely Pen BT = Pen W BT + Pen E BT + Pen S BT + Pen N BT, where Pen W BT = (I 3 (Px 1 E 0 I M ))Σ W (H W V G W ), Pen E BT = (I 3 (Px 1 E N I M ))Σ E (H E V G E ), Pen S BT = (I 3 (I N Py 1 E 0 ))Σ S (H S (58) V G S ), Pen N BT = (I 3 (I N Py 1 E M ))Σ N (H N V G N ). The matrices H W,E,S,N are the discrete boundary operators related to H in (6) and G W,E,S,N are vectors containing the boundary data at the appropriate boundary points. Finally, Σ W,E,S,N are penalty matrices to be determined The semi-discrete energy estimate The discrete energy method applied to (56) (multiplying the equation from the left by V T (I 3 P x P y ) and adding its transpose) and the SBP properties in (54) yield d dt V 2 P +Diss = BT+V T (I 3 P x P y )Pen BT +(V T (I 3 P x P y )Pen BT ) T, (59) where Diss = 2ɛ( (I 3 D x )V 2 P + (I 3 D y )V 2 P ). As in (8), V 2 P = V T (Ĩ P x P y )V defines a discrete semi-norm such that (59) represents the discrete energy rate of the velocity field (u, v) T. The notation BT stands for the discrete boundary term corresponding to (9) in the continuous case. It contains the four contributions from each boundary of the domain, i.e. BT = BT W + BT E + BT S + BT N, where BT W = V T (I 3 E 0 P y )AV + 2ɛV T (I 3 E 0 P y )(Ĩ D x)v, BT E = V T (I 3 E N P y )AV + 2ɛV T (I 3 E N P y )(Ĩ D x)v, (60) BT S = V T (I 3 P x E 0 )BV + 2ɛV T (I 3 P x E 0 )(Ĩ D y)v, BT N = V T (I 3 P x E M )BV + 2ɛV T (I 3 P x E M )(Ĩ D y)v. Following the continuous analysis, we will rewrite each term in BT as a quadratic form similar to (11). First, we project V onto the boundaries by 19

20 V b = B b V, where E 0 I M B b E = N I M I N E 0 I N E M on the west boundary, on the east boundary, on the south boundary, on the north boundary (61) and b W, E, S, N. Next, the discrete analogue of (10) is u n b n b xi NM n b yi NM 0 u s b w b = p b ɛd n u n b = n b yi NM n b xi NM 0 Tb V b, T b = 0 0 I NM ɛd n bn b x ɛd n bn b y 0, (62) ɛd n u s b ɛd n bn b y ɛd n bn b x 0 where D n b = n b xd x + n b yd y approximates the normal derivative. By applying (62) and rearranging, each term in (60) can be written D(u n b) 0 I NM I NM 0 0 D(u n b) 0 0 I NM BT b = (w b ) T (I 5 P b ) I NM w b. I NM } 0 I NM 0 {{ 0 0 } A b n (63) Here, P b is an operator which approximates a line-integral along boundary b, namely E 0 P y on the west boundary, P b E = N P y on the east boundary, (64) P x E 0 on the south boundary, P x E M on the north boundary. Remark 6.1. All matrices in (63) are diagonal, which implies that we are dealing with N M (the total number of grid points) decoupled quadratic forms, each associated to a grid point. The operator P b projects the variables onto boundary b and removes the contributions from the internal grid points. Hence, the number of non-zero terms in (63) is equal to the number of boundary points. 20

21 Remark 6.1 implies that (63) can be rewritten as BT b = (w b ) T (I 5 P b )A n bw b = l b (w l ) T P b ll(a n b) l w l, (65) where l b indicates the set of points belonging to the boundary b W, E, S, N. In (65), w l = (w b ) l is the variable at a specific boundary point, Pll b is the corresponding diagonal element in P b and (A n b) l is the pointwise version of A n in (11) The discrete boundary conditions We derive the discrete boundary condition by replicating step-by-step the continuous procedure in 2.2, but limit ourselves to the weak implementation of the characteristic variable formulation The discrete diagonalization Since all the matrices (A n b) l in (65) have exactly the same structure as A n in (11), they can be diagonalized by the eigenvalue decomposition already derived in Section The negative and positive eigenvalues from each (A n b) l define the vectors λ b 1,2 and λ b 4,5, respectively. Their components are the pointwise version of (17) on boundary b. The block-diagonal discrete version of (21) is D(λ Λ,b = b 1/(2 + (λ b 1) 2 )) 0 0 D(λ b 2/(1 + (λ b 2) 2, (66) )) D(λ Λ +,b = b 4/(1 + (λ b 4) 2 )) 0 0 D(λ b 5/(2 + (λ b 5) 2, (67) )) where the division should be interpreted elementwise. Furthermore, let D(λ (X,b ) T = b 1) 0 I NM I NM 0 0 D(λ b, (68) 2) 0 0 I NM 0 D(λ (X +,b ) T = b 4) 0 0 I NM D(λ b (69) 5) 0 I NM I NM 0 be the block-diagonal version of (27). We define the discrete in- and outgoing characteristic variables as W,b = (X,b ) T w b and W +,b = (X +,b ) T w b, respectively. With this notation, (65) becomes W BT b +,b T Λ = (I 4 P b ) +,b 0 W +,b 0 Λ,b (70) W,b 21 W,b

22 The discrete form of the boundary conditions Recall that the continuous boundary conditions were imposed in the form (22) and, hence, we want to construct boundary operators H b in (58) in a similar way. Consider (68) and (69) and define the boundary operators H +,b = (X +,b ) T T b (I 3 B b ), H,b = (X,b ) T T b (I 3 B b ), (71) with B b as in (61) and T b as in (62). These operators project V onto the boundary b and transform it to the discrete in- and outgoing characteristic variables, W +,b and W,b in (70). Hence, the discrete version of (23) on the boundary b becomes H b V = H,b V RH +,b V = W,b RW +,b (72) where R = [ R I NM ] is the block-diagonal version of R in (22). In particular, the discrete boundary condition at a solid wall (44) is obtained by choosing R as in (47) Stability of the discrete weak implementation By considering (70) and using (72) in (58), the discrete energy rate (59) can be rewritten as d dt V 2 P + Diss = e {W,E,S,N} W +,b T Λ (I 4 P b ) +,b 0 W +,b 0 Λ,b + W,b W,b V T (I 3 P b )Σ b (W,b RW +,b G b )+V T (I 3 P b )Σ b (W,b RW +,b G b )) T (73) which is the discrete analogue of (38). We can now follow the procedure in the continuous analysis and use similar conditions to get a bound. We choose Σ b = (H,b ) T Λ,b, (74) as penalty matrix with Λ,b given in (66), which leads to d dt V 2 P + Diss = (75) W +,b T W,b (I 6 P b ) Λ+,b R T Λ,b 0 W +,b Λ,b R Λ,b Λ,b W,b, e {W,E,S,N} G b 0 Λ,b 0 G b 22

23 which corresponds to (39). By introducing a positive semi-definite matrix Γ b and adopting the splitting in (40), the bound for the discrete energy rate with non-zero data G b becomes d dt V 2 P + 2ɛ ( (Ĩ D x)v 2 P ) + (Ĩ D y)v 2 P (G b ) T (I 2 P b )Γ b G b. b {W,E,S,N} (76) For the characteristic boundary conditions, all the diagonal elements in (66) and (67) remain bounded for all possible values of the components u n b. The conditions on Λ ±,b, R for which a bounded Γ b exists and (76) constitutes a bound are given in Proposition 9. The semi-discrete approximation (56) of (36) with penalty matrix (74) and boundary operators (71)-(72) leads to stability if Proof. See the proof of Proposition 3 and 8. Λ +,b + R T Λ,b R > 0. (77) In the homogeneous cases, the proof of Corollary 2 proves Corollary 3. The semi-discrete approximation (56) of (36) with homogeneous boundary conditions and penalty matrix (74) leads to stability if 7. The fully discrete formulation Λ +,b + R T Λ,b R 0. (78) To advance the divergence relation in time, we use the high order accurate SBP-SAT finite difference technique also in time on (56) and impose the initial condition (7) weakly. To make the analysis self-contained, we shortly introduce this procedure and recommend [29, 22] for more details. Consider a two-dimensional spatial grid with N M points and L time levels. The fully-discrete approximation of a variable v = v(t, x, y) is a vector of length LNM arranged as follows. [. ] v = [v] k, [v] k = v, [ v ]. = ki v kij, where v kij v(t k, x i, y j )... ki 23.

24 Each vector component v k has length NM and represents the discrete variable on the spatial domain at time level k. The SBP approximation of the time derivative is (D t I N I M )v = (Pt 1 Q t I N I M )v v t, where P t and Q t satisfy the same properties as the spatial operators The fully-discrete formulation Consider the fully-discrete variable V = [V 1,..., V k,..., V L ] T, where each V k = (u k, v k, p k ) T represents the variables on the whole spatial domain at the k-th time level. The SBP-SAT approximation of (36) including a weak imposition of the boundary and initial conditions can be written (D t Ĩ3)V + F(V)V = Pen BT + Pen Time. (79) Here, F = blockdiag(f (V 0 ),..., F (V L )) where the blocks are the nonlinear spatial differential operators given by F (V k ) = ( D x ) k + ( D ] y ) k ɛ [(Ĩ D x) 2 + (Ĩ D y) 2, k = 0,..., L. (80) The nonlinearity in (80) is due to the form of ( D x ) k and ( D x ) k given in (57). In (79), Pen BT is a vector of penalty terms for weakly imposing the boundary conditions at each time level, i.e. Pen BT = [(Pen BT ) 0,..., (Pen BT ) k,..., (Pen BT ) L ] T (81) where each (Pen BT ) k is the sum of the four boundary penalties given in (58). Finally, Pen Time is the penalty term for weakly imposing the initial condition given by Pen Time = σ t (Pt 1 E 0 Ĩ I N I M )(V f), (82) where f is a vector of the same length as V containing the initial data at k = 0. Remark 7.1. Note that no initial condition is imposed on the pressure in (82). This is in line with the continuous analysis in Section

25 7.2. The fully discrete energy estimate Consider the diagonal matrix P = (P t I 3 P x P y ) and let v 2 P = v T P v be a discrete L 2 norm with respect to time and space. We can prove Proposition 10. The discretization (79) of (36), with spatial penalty terms (81) satisfying the assumptions of Proposition 9 (or Corollary 3), is stable with the temporal penalty term (82) and σ t = 1. Proof. We apply the discrete energy method to (79) (multiplying the equation from the left by V T P and adding its transpose) to get [ ] V T P (D t + Dt T ) Ĩ3 V + V ( T P F(V) + F T P ) V = (83) + V T P Pen BT + (V T P Pen BT ) T + 2σ t V T 0 (Ĩ P x P y )(V 0 f), with Ĩ3 from (55). By applying the SBP property (54) to the temporal differential operator, the first term on the left-hand side of (83) can be rewritten as [ ] ) V T P (D t + Dt T ) Ĩ3 V = V (B T t Ĩ P x P y V = V L 2 P V 0 2 P, (84) where B t = D( 1, 0,..., 0, 1) and V k 2 P, k = 0, L, is the discrete semi-norm at the first and last time level with respect to space. From (80) and the SBP property (54) applied to the spatial differential operator, the second term in (83) can be expanded into V T (P F(V) +F(V) T P ) V = (85) ( ) 2ɛ (P t Ĩ D x)v 2 P + (P t Ĩ D y)v 2 P BT. Here, BT is the vector BT = [(BT) 0,..., (BT) k,..., (BT) L ] T, where each BT k represents the boundary term on the right-hand side of (59) at time level k. It can be written as the sum of the four contributions coming from each boundary as in (60). With conditions (77) (or (78) in the homogeneous cases) satisfied at each time level, (76) follows and consequently BT + V T P Pen BT + (V T P Pen BT ) T (G b ) T (P t I 2 P b )Γ b G b. 25 e {W,E,S,N} (86)

26 In (86), G b = [ G b 0,..., G b k,..., T L] Gb contains the boundary data G b k on boundary b at time level k, P b is from (64) and Γ b = blockdiag(γ b 0,..., Γ b L ), where each Γ b k is a bounded positive definite matrix. By considering (84), (85) and (86), relation (83) becomes V L 2 P (G b ) T (P t I 3 P b )Γ b G b + V 0 2 P +2σ t V T 0 (Ĩ P x P y )(V 0 f). e {W,E,S,N} Next, we add and subtract f 2 P to (87) which leads to V L 2 P f 2 P + e {W,E,S,N} + [ V0 f (G b ) T (P t I 3 P b )Γ b G b ] T 1 + 2σt σ t (Ĩ σ t 1 P x P y ) (87) V0. (88) f The last term in (88) is negative semi-definite if and only if σ t = 1, which yields V L 2 P f 2 P + (G b ) T (P t I 3 P b )Γ b G b V 0 f 2 P, (89) and we have a bound. e {W,E,S,N} Remark 7.2. Note that (89) is the fully discrete version of the continuous estimate (43) and the semi-discrete estimate (76) with an additional damping term due to the weak imposition of the initial condition. As in the continuous and semi-discrete case, it is a bound for the numerical velocity field only. 8. Conclusions An investigation on the initial boundary value problem for the incompressible nonlinear Navier-Stokes equations have been presented. The velocity-divergence formulation of the problem was chosen to ensure a divergence free solution without additional artificial procedures. The boundary conditions were obtained by considering two different techniques to diagonalize the boundary terms. In particular, a set of non-singular rotations and a standard eigenvalue decomposition lead to the rotated and characteristic formulation, respectively. 26

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